Phase-front-modulation sensor and method of fabrication

ABSTRACT

A sensor and a method of fabrication are provided. The sensor includes at least one optical waveguide and an optical reflector. The optical reflector is optically coupled to the at least one optical waveguide and includes a first portion and a second portion. The first portion is configured to reflect a first portion of light back to the at least one optical waveguide. The second portion is configured to reflect a second portion of light back to the at least one optical waveguide. The reflected second portion of the light differs in phase from the reflected first portion of the light by a phase difference that is not substantially equal to an integer multiple of π when the second portion of the optical reflector is in an equilibrium position in absence of the perturbation.

RELATED APPLICATIONS

This application claims the benefit of priority to U.S. ProvisionalAppl. No. 62/119,647, filed Feb. 23, 2015, which is incorporated in itsentirety by reference herein.

BACKGROUND

1. Field

This application relates generally to sensor systems, and moreparticularly to optical-fiber-compatible acoustic sensor systems.

2. Description of the Related Art

In membrane-based fiber acoustic sensors, a deflectable membrane is usedas a transducer to convert the small vibration induced by an incidentacoustic wave into an optical modulation. See, e.g., M. J. Gander, W. N.MacPherson, J. S. Barton, R. L. Reuben, J. D. C. Jones, R. Stevens, K.S. Chana, S. J. Anderson, and T. V. Jones, “Embedded micromachinedfiber-optic Fabry-Perot pressure sensors in aerodynamics applications,”IEEE Sens. J. 3, 102-107 (2003); L. H. Chen, C. C. Chan, W. Yuan, S. K.Goh, and J. Sun, “High performance chitosan diaphragm-based fiber-opticacoustic sensor,” Sensors Actuators A Phys. 163, 42-47 (2010); J. A.Bucaro, N. Lagakos, and B. H. Houston, “Miniature, high performance,low-cost fiber optic microphone,” J. Acoust. Soc. Am. 118, 1406-1413(2005); F. Xu, J. Shi, K. Gong, H. Li, R. Hui, and B. Yu, “Fiber-opticacoustic pressure sensor based on large-area nanolayer silverdiaphragm,” Opt. Lett. 39, 2838-40 (2014); S. E. U. Lima, O. Frazão, R.G. Farias, F. M. Araúo, L. A. Ferreira, V. Miranda, and J. L. Santos,“Intrinsic and extrinsic fiber Fabry-Perot sensors for acousticdetection in liquids,” Microw. Opt. Technol. Lett. 52, 1129-1134 (2010).

These devices are interesting because they are compact, they can exhibita high sensitivity and a low noise, and they are easily opticallymultiplexed into large arrays. The high sensitivity arises in part fromthe high compliance of sub-micron diaphragms, which will deflect bymeasurable amounts even under a very slight pressure (e.g., about 90nm/Pa for a 450-nm thick square diaphragm, 370 μm on the side; see,e.g., W. Jo, O. C. Akkaya, O. Solgaard, and M. J. F. Digonnet,“Miniature fiber acoustic sensors using a photonic-crystal membrane,”Opt. Fiber Technol. 19, 785-792 (2013)). Because of this unique set offeatures, such devices are being studied and developed for a largenumber of important applications ranging from seismic research (see,e.g., G. Gagliardi, M. Salza, P. Ferraro, P. De Natale, A. Di Maio, S.Carlino, G. De Natale, and E. Boschi, “Design and test of a laser-basedoptical-fiber Bragg-grating accelerometer for seismic applications,”Meas. Sci. Technol. 19, 085306 (2008)) to large structure monitoring(see, e.g., M. Majumder, T. K. Gangopadhyay, A. K. Chakraborty, K.Dasgupta, and D. K. Bhattacharya, “Fibre Bragg gratings in structuralhealth monitoring—Present status and applications,” Sensors Actuators APhys. 147, 150-164 (2008)), underwater surveillance (see, e.g., D. Hilland P. Nash, “Fiber-optic hydrophone array for acoustic surveillance inthe littoral,” in Photonics for Port and Harbor Security, M. J. DeWeertand T. T. Saito, eds., International Society for Optics and Photonics,2005, pp. 1-10), MRI-compatible microphones (see, e.g., M. S. NessAiver,M. Stone, V. Parthasarathy, Y. Kahana, A. Paritsky, and A. Paritsky,“Recording high quality speech during tagged cine-MRI studies using afiber optic microphone,” J. Magn. Reson. Imaging 23, 92-7 (2006)),photoacoustic imaging (see, e.g., P. C. Beard, F. Pérennès, E.Draguioti, and T. N. Mills, “Optical fiber photoacoustic—photothermalprobe,” Opt. Lett. 23, 1235 (1998)), small force measurements (see,e.g., W. Jo and M. J. F. Digonnet, “Piconewton force measurement using ananometric photonic crystal diaphragm,” Opt. Lett. 39, 4533 (2014)),atomic force microscopy (see, e.g., D. Rugar, H. J. Mamin, and P.Guethner, “Improved fiber-optic interferometer for atomic forcemicroscopy,” Appl. Phys. Lett. 55, 2588 (1989)), and bio/chemicalsensors (see, e.g., X.-D. Wang and 0. S. Wolfbeis, “Fiber-optic chemicalsensors and biosensors (2008-2012),” Anal. Chem. 85, 487-508 (2013)).Most of these applications utilize very low minimum detectable pressures(MDPs). For example, for underwater oil and gas exploration, thedetected pressure is typically in the range of 10-200 μPa/√Hz over afrequency that spans from 100 Hz to 20 kHz.

SUMMARY

In certain embodiments, a sensor is provided which comprises at leastone optical waveguide and an optical reflector. The at least one opticalwaveguide is configured to emit light in a direction. The opticalreflector is optically coupled to the at least one optical waveguide,and the optical reflector is configured to reflect at least a portion ofthe light. The optical reflector comprises a first portion and a secondportion. The first portion of the optical reflector is configured toreflect a first portion of the light back to the at least one opticalwaveguide. The second portion of the optical reflector is configured toreflect a second portion of the light back to the at least one opticalwaveguide. The second portion of the optical reflector is responsive toa perturbation by moving relative to the first portion of the opticalreflector. The reflected second portion of the light differs in phasefrom the reflected first portion of the light by a phase difference thatis not substantially equal to an integer multiple of π when the secondportion of the optical reflector is in an equilibrium position inabsence of the perturbation.

In certain embodiments, a method for fabricating a sensor is provided.The method comprises providing a first tube comprising a ferrule insidethe first tube. The ferrule comprises an optical waveguide configured toemit a light beam. The method further comprises inserting at least onelens into the first tube. A portion of the at least one lens extendsoutwardly past an end of the first tube. The at least one lens isconfigured to receive the light beam emitted from the optical waveguide.The method further comprises affixing a first end of a second tube tothe portion of the at least one lens extending outwardly past the end ofthe first tube. The method further comprises affixing an opticalreflector to a surface of a second end of the second tube.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A schematically illustrates an example optical sensor inaccordance with certain embodiments described herein.

FIG. 1B schematically illustrates a cross-section of an opticalreflector in accordance with certain embodiments described herein.

FIG. 1C schematically illustrates example phase fronts of the incidentand reflected beams in accordance with certain embodiments describedherein.

FIG. 1D schematically illustrates another example optical sensor inaccordance with certain embodiments described herein.

FIG. 2A schematically illustrates an example optical reflector inaccordance with certain embodiments described herein.

FIG. 2B schematically illustrates another example optical reflector inaccordance with certain embodiments described herein.

FIG. 3 schematically illustrates an example fabrication process of areflector in accordance with certain embodiments described herein.

FIG. 4 schematically illustrates an example reflector in accordance withcertain embodiments described herein.

FIG. 5A schematically illustrates an example phase-front-modulation(PFM) sensor in accordance with certain embodiments described herein.

FIG. 5B is a photograph of an example PFM sensor in accordance withcertain embodiments described herein.

FIG. 6 is a plot of measured and calculated sensitivity spectra for anexample PFM sensor in accordance with certain embodiments describedherein.

FIG. 7 is a plot of various noise contributions of the example PFMsensor in accordance with certain embodiments described herein.

FIG. 8 is a plot of the measured sensor noise spectrum of the examplePFM sensor in accordance with certain embodiments described herein.

FIG. 9 is a plot of the MDP spectrum of the example PFM sensor inaccordance with certain embodiments described herein compared to ameasured MDP spectrum of a previously-developed FP-based sensor.

FIG. 10A is a plot of a numerically calculated displacement sensitivityand FIG. 10B is a plot of an analytically calculated displacementsensitivity resulting from varying the well depth h₀ and the ratioκ=r_(dia)/w_(col), where r_(dia) and w_(col) are the sensor diaphragmradius and the collimated beam waist, respectively, at an operatingwavelength of 1550 nm.

FIG. 11A illustrates the coupling coefficient at a static condition atrest (no) as a function of well depth h₀ for a fixed value of κ=0.64.

FIG. 11B illustrates the displacement sensitivity σ as a function ofwell depth h₀ for a fixed value of κ=0.64, with the sensitivity going tozero at h₀=λ/4.

FIG. 11C illustrates the displacement sensitivity σ as a function ofwavelength λ for h₀=λ/8 and κ=0.64.

DETAILED DESCRIPTION

Of the several types of optical readout demonstrated in membrane-basedfiber acoustic sensors, the most sensitive are miniature interferometricFabry-Perot (FP) sensors constructed at the tip of an optical fiber(see, e.g., W. Jo, O. C. Akkaya, O. Solgaard, and M. J. F. Digonnet,“Miniature fiber acoustic sensors using a photonic-crystal membrane,”Opt. Fiber Technol. 19, 785-792 (2013); O. Kilic, M. J. F. Digonnet, G.S. Kino, and O. Solgaard, “Miniature photonic-crystal hydrophoneoptimized for ocean acoustics,” J. Acoust. Soc. Am. 129, 1837-1850(2011)). The interferometric FP sensor can comprise a deflectablediaphragm, made reflective by any number of means, and the facingreflective end of a single-mode fiber. When the diaphragm is exposed toan acoustic wave, it vibrates, which modulates the distance between thereflectors and therefore modulates the reflectivity of the FP sensor.This reflectivity modulation can be measured with a laser beam launchedand collected through the fiber. These devices are very sensitivebecause of the aforementioned compliance of thin diaphragms, and becauseas a multiple-wave interferometer, such an interferometric FP sensor canmeasure extremely small displacements (e.g., about 200 fm/√Hz). Smallminimum detectable pressures (MDPs) therefore can be achieved usinghighly compliant diaphragms (e.g., large diameters, small thicknesses)and a large FP finesse (e.g., high mirror reflectivities).

Various implementations of interferometric FP sensors have beenreported, although most references cite either no quantitativesensitivity, or a sensitivity expressed, for example, in V/Pa (which canbe made arbitrarily large with a voltage amplifier) (see, e.g., L. H.Chen, C. C. Chan, W. Yuan, S. K. Goh, and J. Sun, “High performancechitosan diaphragm-based fiber-optic acoustic sensor,” Sensors ActuatorsA Phys. 163, 42-47 (2010); W. Wang, N. Wu, Y. Tian, C. Niezrecki, and X.Wang, “Miniature all-silica optical fiber pressure sensor with anultrathin uniform diaphragm ,” Opt. Express 18, 9006-14 (2010); S. E. U.Lima, O. Frazão, R. G. Farias, F. M. Araújo, L. A. Ferreira, V. Miranda,and J. L. Santos, “Intrinsic and extrinsic fiber Fabry-Perot sensors foracoustic detection in liquids,” Microw. Opt. Technol. Lett. 52,1129-1134 (2010)), which makes comparison impossible. Gander (M. J.Gander, W. N. MacPherson, J. S. Barton, R. L. Reuben, J. D. C. Jones, R.Stevens, K. S. Chana, S. J. Anderson, and T. V. Jones, “Embeddedmicromachined fiber-optic Fabry-Perot pressure sensors in aerodynamicsapplications,” IEEE Sens. J. 3, 102-107 (2003)) used the Fresnelreflections from a microfabricated silicon diaphragm and the cleaved endof a single-mode fiber to form the FP sensor, and achieved an MDP of 900mPa/√Hz. In recent years, our group has improved this figure by severalorders of magnitude by (1) increasing the compliance of the diaphragms,(2) increasing the FP finesse by increasing the reflectivity of thefiber mirror (with a gold or multi-layer dielectric coating), and (3)increasing the diaphragm reflectivity by writing a photonic crystal inthe diaphragm. See, e.g., O. Kilic, M. J. F. Digonnet, G. S. Kino, andO. Solgaard, “Miniature photonic-crystal hydrophone optimized for oceanacoustics,” J. Acoust. Soc. Am. 129, 1837-1850 (2011); O. C. Akkaya, M.J. F. Digonnet, G. S. Kino, and O. Solgaard, “Modeling and demonstrationof thermally stable high-sensitivity reproducible acoustic sensors,” J.Microelectromechanical Syst. 21, 1347-1356 (2012); W. Jo, O. C. Akkaya,O. Solgaard, and M. J. F. Digonnet, “Miniature fiber acoustic sensorsusing a photonic-crystal membrane,” Opt. Fiber Technol. 19, 785-792(2013). The lowest average MDP in air reported using these techniques isabout 2.6 μPa/√Hz between 1 kHz and 30 kHz. In comparison, high-endcommercial capacitive microphones have an MDP of about 0.2 μPa/√Hz overa comparable range (see, e.g., Brüel & Kjær, “Type 4179,”www.bksv.com/Products/transducers/acoustic/microphones/microphone-cartridges/4179).These fiber sensors also operate in water, with an MDP as low as 12μPa/√Hz, which is below sea-state zero, and is comparable to the MDP ofpiezoelectric-based commercial hydrophones (see, e.g., Teledyne Reason,“TC4032,” http://www.teledyne-reson.com/hydrophones/tc-4032/).

Although they exhibit outstanding performance, the FP-based sensorspreviously developed can be time-consuming to fabricate, with multiplesilicate-bonding steps to secure the fiber and diaphragm to a commonsupport. To achieve maximum sensitivity, they also can include eithertuning the laser wavelength within a very narrow range in the vicinityof a resonance of the FP sensor (e.g., ±0.5 nm) or tuning the cavitylength to a high precision during assembly. Achieving a reproduciblecavity length (e.g., for wavelength-multiplexed sensor arrays) ispossible for modest numbers of sensors, but can be unwieldy for verylarge arrays.

Certain embodiments described herein provide a solution in the form of anew type of fiber sensor that utilizes a physical principle new to thisfield and that is referred to herein generally as aphase-front-modulation (PFM) sensor. In contrast to optical modulatorswhich utilize induced phase change via free-carrier dispersion (see,e.g., B. R. Hemenway, O. Solgaard, and D. M. Bloom, “All-siliconintegrated optical modulator for 1.3 μm fiber-optic interconnects,”Appl. Phys. Lett. 55, 349 (1989)), certain embodiments described hereinprovide PFM sensors which, as compared to previously-developed FP-basedsensors, are much easier and faster to assemble, offer a much broaderand reproducible operating wavelength set in the clean room, and havecomparable MDPs for equal diaphragm dimensions.

FIG. 1A schematically illustrates an example sensor 10 in accordancewith certain embodiments described herein. While the example sensor 10is described herein in the context of sensing acoustic pressure waves,various embodiments described herein may more generally be described ascomprising a displacement sensor which is responsive to selectedperturbations (e.g., acoustic pressure waves, magnetic fields, electricfields, accelerations, other forces) by having two or more portionsdisplaced relative to one another and by providing one or more signalsindicative of information regarding one or more aspects of theperturbations (e.g., magnitude, frequency, waveform).

The sensor 10 can comprise at least one optical waveguide 20 configuredto emit light 22 in a direction 24. The sensor 10 can further comprisean optical reflector 30 optically coupled to the at least one opticalwaveguide 20. The optical reflector 30 is configured to reflect at leasta portion of the light 22 and comprises a first portion 32 of theoptical reflector 30 and a second portion 34 of the optical reflector30. The first portion 32 of the optical reflector 30 (e.g., a firstsurface) is configured to reflect a first portion of the light back tothe at least one optical waveguide 20. The second portion 34 of theoptical reflector 30 (e.g., a second surface parallel to a first surfaceof the first portion 32 of the optical reflector 30) is configured toreflect a second portion of the light back to the at least one opticalwaveguide 20. The second portion 34 of the optical reflector 30 isresponsive to a perturbation (e.g., a pressure wave incident on thesensor 10) by moving relative to the first portion 32 of the opticalreflector 30. The second portion 34 of the optical reflector 30 isoffset from the first portion 32 of the optical reflector 30 along thedirection 24 such that the reflected second portion of the light differsin phase from the reflected first portion of the light by a phasedifference that is not substantially equal to an integer multiple of πwhen the second surface 34 is in an equilibrium position in absence ofthe perturbation. For example, a maximum sensitivity can be achieved incertain embodiments in which the phase difference is substantially equalto an odd integer multiple of π/2 (e.g., ±π/2, ±3π/2, ±5π/2). As usedherein, the phrase “substantially equal to” as used in describing thephase difference refers to phase differences that are within ±10% of thecited value (e.g., within ±5% of the cited value, within ±2% of thecited value).

In certain embodiments, the phase difference has a magnitude that is inat least one of the following ranges: substantially greater than zeroand substantially less than n, substantially greater than π andsubstantially less than 2π, and substantially greater than 2π andsubstantially less than 3π. In certain embodiments, the phase differencehas a magnitude that is in at least one of the following ranges:substantially greater than zero and less than or equal to π/2, greaterthan or equal to π/2 and substantially less than π, substantiallygreater than π and less than or equal to 3π/2, greater than or equal to3π/2 and substantially less than 2π, substantially greater than 2π andless than or equal to 5π/2, and greater than or equal to 5π/2 andsubstantially less than 3π. As used herein, the phrase “substantiallygreater than” as used in describing the ranges of the phase differencemagnitude refer to phase difference magnitudes that are greater than102% of the cited value (e.g., greater than 105% of the cited value,greater than 110% of the cited value). As used herein, the phrase“substantially less than” as used in describing the ranges of the phasedifference magnitude refer to phase difference magnitudes that are lessthan 98% of the cited value (e.g., less than 95% of the cited value,less than 90% of the cited value). As used herein, the phrase“substantially greater than zero” as used in describing the ranges ofthe phase difference magnitude refer to phase difference magnitudes thatare greater than π/8. In certain embodiments, the phase difference isselected to provide an amount of sensitivity that is adequate for thepurposes for which the sensor 10 is being used.

In certain embodiments, the at least one optical waveguide 20 comprisesan optical fiber 25. For example, as schematically illustrated by FIG.1D, at least one laser 40 can generate light having at least onewavelength and can be optically coupled to the at least one opticalwaveguide 20 (e.g., comprising a single-mode optical fiber 25) via atleast one optical coupler 42 (e.g., an optical circulator) to providelight having a power level of P_(in) to an input of the at least oneoptical waveguide 20. The at least one optical coupler 42 can be furtheroptically coupled to an optical analyzer 44 (e.g., a dynamic signalanalyzer) such that light having a power level of P_(out) emitted fromthe input of the at least one optical waveguide 20 is provided to theoptical analyzer 44. In certain embodiments, as schematicallyillustrated by FIG. 1D, the optical analyzer 44 comprises an opticalamplifier 46 configured to amplify the light received from the at leastone optical waveguide 20. In certain embodiments, the at least oneoptical waveguide 20 does not comprise an optical fiber 25, but can bepart of an integrated optical circuit that uses the at least one opticalwaveguide 20 to transport input optical signals to the optical reflector30 and to transport reflected optical signals from the optical reflector30 (e.g., to an optical analyzer 44).

In certain embodiments, the at least one optical waveguide 20 canfurther comprise at least one lens 26 (e.g., a graded-index (GRIN) lens)which modifies the spatial configuration of the light 22. The at leastone lens 26 can be configured to receive the light 22 from the opticalfiber 25 and to transmit the light towards the optical resonator 30. Forexample, the optical fiber 25 can transmit the light 22 as a light beamfrom an output of the optical fiber 25, and the at least one lens 26 canbe configured to collimate the light 22 from the optical fiber 25 and toreturn the reflected first and second portions of the light to theoptical fiber 25. In certain embodiments, the optical fiber 25 and theat least one lens 26 are integral with one another forming a unitaryoptical waveguide 20 or a monolithic optical waveguide 20. For example,the at least one lens 26 can be mechanically coupled or otherwiseaffixed to an end of the optical fiber 25. In certain other embodiments,the at least one lens 26 is separate from (e.g., not integral with, notforming a unitary structure with) the optical fiber 25.

An optical reflector 30 (e.g., comprising a semiconductor wafer) can beplaced a short, non-critical distance (e.g., a few millimeters) from theat least one optical waveguide 20. In certain embodiments, the opticalreflector 30 comprises at least one of the following materials: silicon,silicon nitride, silicon carbide, graphene. The reflector 30 cancomprise a first portion 32 that is optically reflective. For example,the first portion 32 can comprise an optically-reflective first surface(e.g., an optical-quality planar surface of the wafer, a metal layer) oran optically-reflective structure (e.g., a photonic crystal structure).

The reflector 30 can further comprise a region 36 bounded at least inpart by the first portion 32. For example, the region 36 can comprise awell microfabricated into the wafer (e.g., having a λ/8 depth and aplanar bottom), as schematically illustrated by FIG. 1B. In certainembodiments, the well can have a perimeter in a plane parallel to thefirst surface of the first portion 32 that is circular, square,rectilinear, triangular, or another shape. In certain embodiments, theperimeter of the well can be surrounded by the first portion 32 or canbe bounded on one, two, or more sides by the first surface of the firstportion 32.

The second portion 34 of the reflector 30 can comprise a diaphragm inthe region 36 (e.g., at the bottom of the well). The diaphragm 38 can besignificantly thinner (e.g., by hundreds of nanometers) than portions ofthe wafer surrounding the diaphragm 38 such that the diaphragm 38 iselastically movable and/or elastically deformable in response to theperturbation. 38. In certain embodiments, the diaphragm 38 can have ashape in a plane parallel to the first surface of the first portion 32that is circular, square, rectilinear, triangular, or another shape. Thediaphragm 38 can comprise a reflective second surface in the region 36(e.g., at the bottom of the well) or an underlying reflective layer inthe region 36 (e.g., beneath a surface of the bottom of the well).

In certain embodiments, the collimated light 22 from the at least oneoptical waveguide 20 can be positioned and can have a width such that afirst portion of the light 22 is incident on the first portion 32 of thereflector 30, while a second portion of the light 22 is incident on thesecond portion 34 of the reflector 30 (e.g., the diaphragm 38 in thewell). In certain embodiments, the ratio of the optical power of thefirst portion of the light 22 to the optical power of the second portionof the light 22 is in a range between 0.3 and 0.7, in a range between0.4 and 0.6, or in a range between 0.45 and 0.55.

In certain embodiments, as schematically illustrated by FIGS. 1A, 1B,and 1D, the light 22 can have a width that is greater than a width ofthe well and can be positioned such that the light 22 irradiates thewhole area of the second portion 34 (e.g., irradiates the whole bottomarea of the well) and irradiates at least some of the first portion 32of the reflector 30. For example, the light 22 can be centered on thewell and the first portion of the light 22 (e.g., an outer portion) canbe incident on the first portion 32 of the reflector outside the well,while the second portion of the light 22 (e.g. an inner or centralportion) of the light 22 is incident on the second portion 34 of thereflector 30 (e.g., the diaphragm 38 in the well). In certain otherembodiments, the light 22 can have a width that is less than or equal toa width of the well and can be positioned such that the light 22irradiates at least a portion of the perimeter of the well such that afirst portion of the light 22 is incident on the first portion 32 of thereflector 30 outside the well, while a second portion of the light isincident on the second portion 34 of the reflector 30 (e.g., thediaphragm 38 in the well).

The light reflected from the first portion 32 of the reflector 30 andthe second portion 34 of the reflector 30 can form a reflected beam oflight which is returned to the at least one optical waveguide 20 (e.g.,via the at least one lens 26). In certain embodiments, the ratio of theoptical power of the reflected first portion of the light 22 to theoptical power of the reflected second portion of the light 22 is in arange between 0.3 and 0.7, in a range between 0.4 and 0.6, or in a rangebetween 0.45 and 0.55.

In certain embodiments, the depth of the well (e.g., one-eighth of thewavelength of the light, which can be expressed as λ/8) can be selectedsuch that after reflection in the absence of a perturbation (e.g., in anequilibrium position of the diaphragm 38), the two reflected portionsare in quadrature, as schematically illustrated by FIG. 1C. Thereflector 30 can be oriented perpendicular to the direction 24 of thelight 22 so that the reflected beam is sent back through the at leastone lens 26 (e.g., the GRIN lens) and is focused onto the core of theoptical fiber 25 of the at least one optical waveguide 20. Because ofthe intentional phase mismatch between the inner and outer portions ofthe reflected beam, after the reflected beam is focused back to the atleast one optical waveguide 20 (e.g., to the core of the optical fiber25 via the at least one lens 26 at the output of the at least oneoptical waveguide 20), it does not couple well (e.g., P_(out) /P_(in)equals about 50%) into the single-mode fiber core. The returning opticalpower from the input port of the at least one optical waveguide 20(e.g., measured by the optical analyzer 44) can then be low.

When a perturbation (e.g., a static acoustic pressure) is incident onthe reflector 30, the diaphragm 38 can be displaced with respect to thethicker outer portion of the reflector 30. This displacement from theequilibrium position of the diaphragm 38 can modify the relative phasebetween the inner and outer portions of the reflected beam, whichchanges the coupling efficiency of the reflected beam to the at leastone optical waveguide 20 (e.g., into the core of the optical fiber 25),and thus changing the optical power returning from the sensor 10, asschematically illustrated by FIG. 1D. A measurement of this power changecan provide the magnitude of the perturbation (e.g., the value of theapplied pressure). The principle is the same for a dynamic pressure atfrequency f_(a). The diaphragm 38 then vibrates at frequency f_(a), thereturning signal is modulated at f_(a), and the measurement can provideboth the amplitude and frequency of the pressure wave.

In certain embodiments, the depth of the well of the region 36 of thereflector 30 is selected to provide the preselected phase differencebetween the two reflected portions. Selection of the depth of the wellcan advantageously provide a relatively simple fabrication for thesensor 10. However, in certain other embodiments, other structuralattributes (e.g. materials, structures) of the region 36 and of thefirst portion 32 of the reflector 30 can be selected to provide thepreselected phase difference between the two reflected portions. Forexample, one or both of the region 36 and the first portion 32 can havean appropriate photonic crystal structure, including appropriatematerials, to provide the preselected phase difference between lightreflected from the region 36 and light reflected from the first portion32.

An example reflector 30 in accordance with certain embodiments describedherein was fabricated at Stanford Nanofabrication Facility on a 4-inchsilicon-on-insulator (SOI) wafer with a 2-μm device layer and a buriedoxide layer. Fabrication utilized conventional clean-room techniques andequipment. A sensor 10 of certain such embodiments can be morestraightforward and faster to fabricate than photonic-crystal FP sensorheads because of its simplicity and greater tolerance on physicaldimensions. The top surface of the fabricated phase plate was coatedwith a 7-nm chromium adhesion layer and a 15-nm gold layer to increaseits power reflectivity to a measured value of about 70%.

FIGS. 2A and 2B schematically illustrate two example reflectors 30A, 30B(e.g., formed from a silicon-on-insulator wafer) in accordance withcertain embodiments described herein. In FIG. 2A, the example reflector30A is schematically shown in a top view (the top portion of FIG. 2A)and in a cross-sectional view (the bottom portion of FIG. 2A) along thedashed line in the top portion of FIG. 2A. The example reflector 30Acomprises a movable portion 50 (e.g., a diaphragm 38 of the secondportion 34 of the reflector 30), a non-movable portion 52 comprising thefirst portion 32 of the reflector 30, and a plurality of springstructures 54 mechanically coupled to the movable portion 50 and to thenon-movable portion 52. While the example reflector 30A of FIG. 2A haseight spring structures 54, other reflectors 30A with 2, 3, 4, 5, 6, 7,9, 10, or more spring structures 54 are also compatible with certainembodiments described herein. The movable portion 50 and the pluralityof spring structures 54 can be defined (e.g., separated from thenon-movable portion 52) by a plurality of gaps 56 cut into the examplereflector 30A (e.g., using microfabrication techniques). The examplereflector 30A is configured to have the movable portion 50 vibrate bytranslating in a direction 58 that is generally perpendicular to themovable portion 50 (e.g., the diaphragm 38) while the movable portion 50generally retains its shape (e.g., planar), and while the springstructures 54 elastically stretch and move.

In FIG. 2B, another example reflector 30B is schematically shown in atop view (the top portion of FIG. 2B) and in a cross-sectional view (thebottom portion of FIG. 2B) along the dashed line in the top portion ofFIG. 2B. The example reflector 30B comprises a movable portion 60 (e.g.,a diaphragm 38 comprising the second portion 34 of the reflector 30) anda non-movable portion 62 comprising the first portion 32 of thereflector 30. The diaphragm 38 can be mechanically coupled to the firstportion 32 of the reflector 30 along a perimeter of the diaphragm 30(e.g., by an edge 64 of the non-movable portion 62 that is mechanicallycoupled to the movable portion 60). The example reflector 30B isconfigured to have the movable portion 60 vibrate, not by translating,but by elastically bowing back and forth (e.g., elastically stretchingand moving) in a direction 68 that is generally perpendicular to themovable portion 60 (e.g., the diaphragm 38), thereby changing the shapeof the movable portion 60. When not vibrating, the movable portion 60 ofthe example reflector 30B can have a planar shape.

In certain embodiments, since the movable portion 50 of the examplereflector 30A generally translates without distortion of its shape, theexample reflector 30A can advantageously provide freedom to select asize and thickness of the movable portion 50 independently of thestress-responsive properties of the movable portion 50. Thus, anacoustic sensor 10 utilizing the example reflector 30A in certainembodiments can be optimized for the use of various beam sizes andpressure levels. In certain embodiments, the example reflector 30B has asimpler fabrication process than does the example reflector 30A, howeverit has a lower optical sensitivity because the movable portion 60 of theexample reflector 30B bows instead of merely translating in a directiongenerally perpendicular to its surface.

FIG. 3 schematically illustrates an example fabrication process of areflector 30 in accordance with certain embodiments described herein. Incertain embodiments, the fabrication process of the reflector 30involves two steps of device thinning by thermal oxidation following adry etching process (e.g., to define spring structures 54 and to releasethe movable portion 50 of the example reflector 30A). The examplefabrication process of FIG. 3 is compatible with fabricating the examplereflector 30A (e.g., including defining the spring structures 54) andwith fabricating the example reflector 30B (e.g., not defining thespring structures 54 around the movable portion 60). Note that theprocess shows the fabrication process of an example reflector 30A. Theexample reflectors 30B were fabricated using the same process except thespring structures 54 were not defined around the sensor diaphragm 38.

FIG. 4 schematically illustrates an example reflector 30A with fourspring structures 54 in accordance with certain embodiments describedherein. The left side of FIG. 4 shows a cross-section of the examplereflector 30A and the right side of FIG. 4 shows a top view of theexample reflector 30A. The perforations 70 along the left portion of thestructure shown on the right side of FIG. 4 can be used to separate theexample reflector 30A from the surrounding portion of the wafer at theappropriate stage of fabrication. The example reflector 30A can havevarious sizes and shapes in accordance with certain embodimentsdescribed herein. In FIG. 4, a generally circular diaphragm 38 comprisesthe moving portion 50 that is mechanically coupled to the surroundingnon-moving portion 52 (e.g., the surrounding portions of the wafer) by aplurality of spring structures 54 (e.g., four elongate structures thatare configured to distort elastically such that the diaphragm 38translates in a direction that is generally perpendicular to the planeof the diaphragm 38). Various sizes of reflectors 30A, diaphragms 38,and spring structures 54 with various compliances can be used. Forexample, two sizes of diaphragms 38 (with radius a=120 μm, 140 μm) with5 different compliances (spring constants) can be fabricated at the sametime. Other sizes and shapes of the diaphragm 38 and the springstructures 54, number of spring structures 54, arrangements of springstructures 54, compliance of the spring structures 54 may be used inaccordance with certain embodiments described herein.

For an example reflector 30B in which the diaphragm 38 is configured tobow (e.g., using a movable portion 60), diaphragms 38 of various shapes,sizes, and thicknesses can be used. For example, a reflector 30B cancomprise a generally circular diaphragm 38 with a thickness in a rangebetween 0.3 μm to 1.5 μm (e.g., 0.45 μm, 1.1 μm) with a radius a=100 μm,110 μm, 120 μm, 130 μm, 140 μm, 150 μm, 160 μm, 170 μm, 180 μm, or 190μm. Other structural parameters of the reflector 30B may be used inaccordance with certain embodiments described herein.

There are many possible modifications that can be made to improve thesensitivity and usability of the PFM sensor for different applicationsin accordance with certain embodiments described herein. For example,different types of reflectors 30 can be implemented at the sensorsurface. It can be desirable to maximize the reflectivity of the sensorsurface because the sensitivity is proportional to the reflectivity. Inthe example PFM sensor that was fabricated, a gold coating was used toachieve reflectivity of about 70%. The reflectivity can be increasedusing different high-reflection coatings such as silver, aluminum, anddielectrics. Also, a photonic crystal bandgap structure could befabricated for the sensor diaphragm 38 and the rest of sensor surface,which can increase its reflectivity (e.g., up to about 99%). For anotherexample, the diaphragm shape is not limited to a circular shape, and itcan be fabricated in other desirable shapes (e.g., rectangular). Incertain embodiments, the spring structures 54 can be simple elongatestructures that are configured to stretch, while in certain otherembodiments, other spring structures 54 (e.g., spiral springs, foldedsprings) can be used to suspend the movable portion 50 (e.g., the sensordiaphragm 38). In certain embodiments, the at least one lens 26 of thesensor 10 schematically illustrated by FIG. 1D (e.g., the GRIN lens) canbe replaced with one or more lenses configured to collimate a laser beamto a target mode size.

FIG. 5A schematically illustrates an example PFM sensor 10 in accordancewith certain embodiments described herein and FIG. 5B is a photograph ofan example PFM sensor 10 in accordance with certain embodimentsdescribed herein. In certain embodiments, the at least one opticalwaveguide 20 comprises an optical fiber 25 and at least one lens 26(e.g., a collimator, which can include a commercially-available0.23-pitch GRIN lens), at least one tube 70, and a fiber-pigtailedferrule 76 mechanically coupled to the at least one optical waveguide20, as schematically illustrated by FIG. 5A. In certain embodiments, theat least one tube 70 and the ferrule 76 can both be made of glass.

In certain embodiments, an example method for fabricating an examplesensor comprises providing a first tube 72 comprising a ferrule 76inside the first tube 72, the ferrule 76 comprising an optical waveguide25 configured to emit a light beam. The example method further comprisesinserting at least one lens 26 into the first tube 72, a portion of theat least one lens 26 extending outwardly past an end of the first tube72, the at least one lens 26 configured to receive the light beamemitted from the optical waveguide 25. The example method furthercomprises affixing a first end of a second tube 74 to the portion of theat least one lens 26 extending outwardly past the end of the first tube72. The example method further comprises affixing an optical reflector30 to a surface of a second end of the second tube 74.

During an example fabrication process in accordance with certainembodiments described herein, the at least one tube 70 comprises a firsttube 72 and a second tube 74. The at least one lens 26 (e.g., GRIN lens)and the ferrule 76 can be inserted inside the first tube 72, with aportion of the at least one lens 26 extending outwardly past an end ofthe first tube 72. The distance 78 between the ferrule 76 and the atleast one lens 26 can be adjusted to modify the radius of the collimatedbeam propagating from the at least one lens 26. For example, the sensor10 shown in FIG. 5B can be fabricated such that the collimated beamemitted from the at least one lens 26 extending from the end of thefirst tube 72 can have a radius of about 630 μm (e.g., an optimum sizefor a diaphragm of radius a=380 μm) at the planned location of thediaphragm 38 (e.g., 8 mm away from the output face of the at least onelens 26). Once the target beam size is achieved (e.g., measured using amechanical chopper), the ferrule 76 and the at least one lens 26 can bebonded to the first tube 72 (e.g., using an epoxy).

In certain embodiments, the second tube 74 can then be affixed (e.g.,using epoxy) to the portion of the at least one lens 26 extendingoutwardly past the end of the first tube 72. This intermediate assemblycan then be held vertically and the reflector 30 (e.g., comprising asilicon structure) can be placed at the top of the second tube 74. Thisexample method can ensure that the angular alignment of the beam withrespect to the reflector 30 is determined by the parallelism of thesurface of the reflector 30 and the end of the second tube 74 (e.g.,which can be polished perpendicular to the longitudinal axis of thesecond tube 74 within ±0.2 degree). Once the diaphragm 38 and beam arewell aligned, as determined by a measurement of the returning power, asmall weight can be placed on top of the silicon phase plate of thereflector 30, and epoxy can be applied to bond the phase plate of thereflector 30 to the second tube 74.

Prior to assembly, at least one vent hole 80 can be made (e.g., drilled)through the side of the second tube 74, as schematically illustrated byFIG. 5A. In certain such embodiments, the at least one vent hole 80 canequalize the pressure of the closed volume of the medium (e.g., air, orwater in the case of a hydrophone) contained in the second tube 74 andthe pressure outside the second tube 74. Without this at least one venthole 80, variations in either the temperature of the contained volume ofthe medium or in the outside static pressure may displace the diaphragm38 and alter the sensitivity.

The normalized sensitivity S_(N) of an acoustic sensor can be defined asthe change in reflected power for a given incident pressure, andnormalized to the laser power incident on the sensor. The normalizedsensitivity S_(N) is then equal to the product of the differentialpressure that develops across the diaphragm (acoustic response R_(a)),the flexibility of the diaphragm (mechanical compliance C_(m)), and theoptical sensitivity (displacement sensitivity σ). The acoustic responsecan be calculated using the same approaches as reported previously (W.Jo, O. C. Akkaya, O. Solgaard, and M. J. F. Digonnet, “Miniature fiberacoustic sensors using a photonic-crystal membrane,” Opt. Fiber Technol.19, 785-792 (2013)), and the mechanical compliance (e.g., of thecircular diaphragm) can be expressed using the closed-form expressionderived previously (S. P. Timoshenko and S. Woinowsky-Krieger, Theory ofPlates and Shells (McGraw-Hill, 1959)). The displacement sensitivity σcan be determined from the coupling coefficient η of the reflected beaminto the waveguide (e.g., the core of the single-mode fiber):

η=2π∫₀ ^(∞) E ₀*(r)E _(r)(r)e ^(iφ(r)) rdr| ²   (1)

where E₀(r) is the electric field of the fiber LP₀₁ mode (normalized toa power of unity), r is the radial direction in the fiber's cylindricalcoordinates (e.g., as defined in FIG. 1B), and E_(r)(r)e^(iφ(r)) is thenormalized electric field of the focused reflected beam at the end ofthe fiber. The partial phase-front modulation imparted to the reflectedfield by the well in the diaphragm can be expressed as:

$\begin{matrix}{{{\varphi (r)} = 0}{for}{r \geq a}{{\varphi (r)} = {{2\frac{2{\pi \left( {h_{0} + {h(r)}} \right)}}{\lambda}} = {\varphi_{b} + {{\delta\varphi}(r)}}}}{for}{r < a}} & (2)\end{matrix}$

where a and h₀ are the radius and depth of the well, respectively, λ isthe optical wavelength, and h(r) is the small diaphragm displacementimparted by an incident acoustic pressure. The term φ_(b)=4πh₀/λ is thephase difference between the inner and outer portions of the reflectedbeam (e.g., the phase bias of the two-wave interferometer). The secondterm δφ(r)=4πh(r)/λ is the small perturbation of this phase differenceinduced by the incident pressure.

When exposed to a pressure, a rigidly displaced diaphragm 38 (e.g., asschematically illustrated by FIG. 2A) moves in a direction perpendicularto the diaphragm 38 without substantially bowing, such that neither hnor δφ are functions of the radial position r. When exposed to apressure, a bowing diaphragm 38 (e.g., as schematically illustrated byFIG. 2B) does not move solely perpendicularly to the diaphragm 38 butactually bows, and therefore both h and δφ are functions of the radialposition r. To accurately calculate the spatial dependence of the phaseshift across a reflective bowing diaphragm 38, and thus the couplingcoefficient and its dependence on applied pressure, the actual profileh(r) of the bowing diaphragm 38 can be taken into account. For ahomogeneous circular membrane with constrained motion on its perimeter,this profile can be expressed as:

$\begin{matrix}{{h(r)} = {h_{m}\left( {1 - \frac{r^{2}}{a^{2}}} \right)}^{2}} & (3)\end{matrix}$

where h_(m) is the peak displacement amplitude, proportional to theapplied pressure. The displacement sensitivity is related to thecoupling coefficient by:

$\begin{matrix}{\sigma = {\lim\limits_{h_{m}\rightarrow 0}\frac{\eta}{h_{m}}}} & (4)\end{matrix}$

The coupling coefficient (Eq. 1) and the displacement sensitivity (Eq.4) can be calculated numerically. However, a closed-form expression ofthe sensitivity can also be derived to gain insight into the devicebehavior. With straightforward manipulation, to first order in theperturbation h_(m), Equation 1 can be rewritten as:

$\begin{matrix}{\eta = {I_{1}^{2} + I_{2}^{2} + {2I_{1}I_{2}{\cos \left( \varphi_{b} \right)}} - {\frac{8\pi}{\lambda}I_{2}I_{3}{\sin \left( \varphi_{b} \right)}h_{m}}}} & (5)\end{matrix}$

where I₁, I₂ and I₃ are overlap integrals involving the reflected field,the LP₀₁ fiber mode field. In the case of I₃, the r-dependentdisplacement of the diaphragm can be expressed as:

$\begin{matrix}{I_{1} = {2\pi {\int\limits_{0}^{a}{{E_{0}^{*}(r)}{E_{r}(r)}r{r}}}}} & \left( {6a} \right) \\{I_{2} = {2\pi {\int\limits_{a}^{\infty}{{E_{0}^{*}(r)}{E_{r}(r)}r{r}}}}} & \left( {6b} \right) \\{I_{3} = {2\pi {\int\limits_{0}^{a}{{E_{0}^{*}(r)}{E_{r}(r)}\left( {1 - \frac{r^{2}}{a^{2}}} \right)r{r}}}}} & \left( {6c} \right)\end{matrix}$

For a given diaphragm radius a, collimated beam radius W, and fiber modespot size, I₁ and I₂ have a fixed numerical value independent of theamplitude h_(m) of the displacement. The perturbation (diaphragmdisplacement) is contained entirely in I₃.

Equation 5 has the classical form of the response of a two-waveinterferometer. The displacement sensitivity (Eq. 4) can be easilycalculated by taking its derivative with respect to h_(m), which gives:

$\begin{matrix}{\sigma = {{- \frac{8\pi}{\lambda}}I_{2}I_{3}{\sin \left( \varphi_{b} \right)}}} & (7)\end{matrix}$

This result provides a simple expression for displacement sensitivity ofan example phase-front modulation sensor. It states in particular thatthe sensor responds linearly to a small displacement (and therefore to apressure), and that the displacement sensitivity is maximum when thephase bias is φ_(b)=4πh₀/λ=π/2, as expected for a two-waveinterferometer, or equivalently a well depth h₀=λ/8, as stated herein.The analytical expression of the displacement sensitivity (Eq. 7) agreeswell with numerical evaluation of Equation 1 over a wide range of valuesof the beam and well radii.

This result shows, and simulations concur, that the displacementsensitivity for this example PFM sensor, in accordance with certainembodiments described herein, is maximum when four conditions are met,namely: (1) the well depth is λ/8 (independently of the relative sizesof the diaphragm and incident beam), (2) the diaphragm radius is 64% ofthe beam radius (which maximizes the product I₂I₃ in Eq. 7), (3) thereflectivity of the phase plate is 100% (which maximizes the amplitudeof the reflected field E_(r)(r) in the integrals of Eqs. 6a-6c), and (4)the phase plate is normal to the incident beam.

Being a two-wave interferometer, unlike previously-developed FP-basedacoustic sensors, the PFM sensor of certain embodiments described hereincan be advantageously much less sensitive to the choice of interrogatingwavelength. It can easily be shown with Eq. 7 that for a well depth ofλ/8, the wavelength can be detuned from its optimum value over a rangeof about ±λ/10 nm before the sensitivity decreases from its maximumvalue by 10%. This is a drastic improvement over the limited bandwidth(e.g., ±0.5 nm) of previously-developed FP-based sensors.

The fabricated phase plate compatible with certain embodiments describedherein had a measured diaphragm radius of 310 μm, a thickness of 1.1 μm(measured with a scanning electron microscope), and a well depth ofabout 500 nm (measured with an optical profilometer). This last value islarger than the optimum targeted value (λ/8≈194 nm for operation atλ=1550 nm) because the diaphragm buckled during microfabrication.Because of the residual angular misalignment between the phase plate andthe collimated beam, only half of the expected power was coupled back tothe fiber, which reduced the displacement sensitivity by a factor of 2.

Using these measured values in the models described herein give acalculated compliance C_(m) of about 8 nm/Pa, a calculated displacementsensitivity σ=3.45×10⁵ m⁻¹ (79% of what it would have been with a welldepth of 194 nm) Multiplying these two values by the calculated acousticresponse R_(a) (the only one of these three parameters that depends onacoustic frequency) gives the predicted normalized sensitivity spectrumshown in FIG. 6 (dotted curve; labeled “Calculated”). The sensitivity isuniform between about 100 Hz and about 10 kHz. In this flat band, thenormalized sensitivity is S_(N)=2.8×10⁻³ Pa⁻¹. Note that this value issmaller than that in a previously-developed FP-based sensor (about 0.17Pa⁻¹), mainly because (1) this new sensor is a two-wave interferometer,(2) its diaphragm is both smaller and thicker, and therefore much lesscompliant (by a factor of about 11), and (3) the well depth (about 500nm) is larger than the ideal value (e.g., 196 nm) and thus u is notoptimum. The dashed spectrum in FIG. 6 (labeled “Targeted”) is thesensitivity that can be expected if all four conditions had been met.The various fabrication errors and misalignments resulted in a factor ofabout 4.6 reduction in the expected sensitivity.

The normalized sensitivity and noise of the example PFM sensor inaccordance with certain embodiments described herein were characterizedexperimentally in an anechoic chamber using a setup similar to the onedescribed previously (W. Jo, O. C. Akkaya, O. Solgaard, and M. J. F.Digonnet, “Miniature fiber acoustic sensors using a photonic-crystalmembrane,” Opt. Fiber Technol. 19, 785-792 (2013)). A new acousticsource (FOSTEX FF85WK) with a wide bandwidth (100 Hz-30 kHz) wasinstalled in the chamber to measure the sensor response down to 100 Hz.The signals from both the sensor and a calibrated reference microphone(Brüel & Kjær 4113) were recorded simultaneously using a dynamic signalanalyzer (HP 3562A). These signals showed a strong coherence (about 1)over the entire frequency range of measurements (100 Hz to 30 kHz). Tomeasure the sensitivity, the sensor signal was calibrated against thereference signal, then normalized to the input power P_(in).

The measured sensitivity spectrum is displayed in FIG. 6 as the solidcurve (labeled “Measured”). As predicted by the “Calculated” curve, itis fairly uniform over a broad band extending from 100 Hz to about 10kHz. The measured sensitivity in the geometric middle of this flat band(1 kHz) is about 2.4×10⁻³ Pa⁻¹, which is in good agreement with thevalue predicted for this example PFM sensor in accordance with certainembodiments described herein (“Calculated” curve in FIG. 6). The smallsensitivity difference in the flat band is likely due to a slightdeparture between the actual radius of the beam incident on thediaphragm and the ideal radius (about 630 μm) for this diaphragm radius.

To find the optimum optical power to achieve the lowest minimumdetectable pressure (MDP), the total noise equivalent power of thedetected signal was calculated by adding the various noise contributionsevaluated at 1 kHz for a detected power P_(det) in the range of 0.01-10mW (see FIG. 7). The noise equivalent power of the detector (4.9 pW/√Hz)was obtained from the manufacturer's datasheet. The laser intensitynoise was calculated from the known RIN (−140 dB/Hz at 1 kHz) of thelaser times the detected power. The thermomechanical noise of thediaphragm was calculated using an equivalent electromechanical circuitmodel reported elsewhere (O. C. Akkaya, M. J. F. Digonnet, G. S. Kino,and O. Solgaard, “Modeling and demonstration of thermally stablehigh-sensitivity reproducible acoustic sensors,” J.Microelectromechanical Syst. 21, 1347-1356 (2012)). FIG. 7 indicatesthat for detected powers lower than about 20 μW, the total noise isdominated by detector noise. Above about 100 μW, the total noise isdominated by the laser intensity noise, and it increases proportional tothe detected power. The shot noise (both optical and electrical) and thethermomechanical noise are negligible at all power levels. Thethermomechanical noise is negligible, unlike in the previously-developedFP-based sensor where it dominates, for the same reason that thesensitivity is lower in this example PFM sensor, which is that thisexample PFM sensor has a much lower finesse since it is a two-waveinterferometer.

The MDP p_(min) of the example PFM sensor is, by definition, thepressure that induces an output signal equal to the noise powerP_(noise), which can be expressed as:

$\begin{matrix}{p_{\min} = \frac{P_{noise}}{S_{N}P_{in}}} & (8)\end{matrix}$

where P_(in) is the laser power incident on the example PFM sensor. Thissignal P_(in) is proportional to the detected power, and so is the noisepower above about 100 μW of detected power. Therefore, at largerdetected powers (>0.7 mW), the MDP is independent of detected power andis at its lowest values. Thus, to achieve the lowest MDP, the examplePFM sensor was operated at 1 mW of detected power.

At 1 kHz, the calculated sensor noise for the example PFM sensor isabout 0.1 nW/√Hz, which is about 10 times lower than the noise reportedfor the previously-developed FP-based sensor. This is due to theelimination in the example PFM sensor of the thermomechanical noise thatdominated in the previously-developed FP-based sensor.

The noise power spectral density of the example PFM sensor was measuredwith the acoustic source turned off (solid curve in FIG. 8). At 1 kHz,it is in good agreement with the laser intensity noise provided by themanufacturer (black circle in FIG. 8). To verify that the dominant noisesource did not originate from the example PFM sensor, the noisemeasurement was repeated after replacing the example PFM sensor by astationary reflector (a dielectric-coated fiber) with the same detectedpower (1 mW). Because the stationary mirror does not respond to acousticperturbations, the noise was expected to have the same contributions asthe actual example PFM sensor, excluding the thermo-mechanical noise.The measured noise using the stationary mirror is indeed almostidentical to the measured sensor noise (see FIG. 8), and both aredominated by laser intensity noise, as expected.

The MDP was calculated by dividing the noise power spectral density bythe calibrated sensor response (in V/Pa). FIG. 9 shows that the lowestmeasured MDP for the example PFM sensor is about 2 μPa√Hz at about 27kHz, and that the average MDP is about 5.4 μPa√Hz between 1 kHz and 30kHz. At 1 kHz, the MDP is about 4 times higher than that of the bestpreviously-developed FP-based sensor (about 4 μPa√Hz). This is becausein the example PFM sensor the noise is 10 times smaller, the normalizedsensitivity about 68 times smaller, and the input power is about 2 timeslarger. Inserting these values in Equation 2 shows that its MDP wasexpected to be smaller by a factor of 68/10/2=3.4, which is consistentwith the Measured factor of about 4.

Near 10 kHz, the MDPs of both sensors are comparable because (1) theexample PFM sensor has an even lower noise (by a factor of 2) becausethe RIN is smaller at high frequencies, and thus the MDP is lower by afactor of 2; and (2) the previously-developed FP-based sensor has ahigher thermomechanical noise (by a factor of 2) due to its resonancearound that frequency, and thus its MDP is higher by a factor of 2.

The example PFM sensor can be further optimized in a number ofstraightforward ways to improve its sensitivity and lower its MDP. Byreducing the diaphragm thickness from the current value of 1.1 μm to thesame thickness as the previously-developed FP-based sensor (e.g., 450nm), C_(m) can be increased by a factor of about 14 (since it isinversely proportional to the third power of thickness). Optimizing thedepth of the well to the targeted value (e.g., λ/8) can increase thedisplacement sensitivity by a factor of about 1.3. The reflectivity ofthe phase plate can also be increased from about 70% to nearly 100%, forexample, by writing in it a photonic crystal. These combinedimprovements are predicted to yield an MDP at 1 kHz of about 0.59μPa/√Hz, which is lower than the value reported for the bestpreviously-reported FP-based sensor (dotted spectrum in FIG. 9; labeled“Measured”). Note that improving the optical alignment would increasethe sensor output signal and hence the normalized sensitivity. However,it would reduce the sensor noise as well, in the same ratio as long asthe noise is limited by the laser RIN, and it would therefore notimprove the MDP.

As described herein, four conditions can be considered for achieving amaximum sensitivity in accordance with certain embodiments describedherein, namely: (1) the well depth is λ/8 (independently of the relativesizes of the diaphragm and incident beam), (2) the diaphragm radius is64% of the beam radius (which maximizes the product I₂I₃ in Eq. 7), (3)the reflectivity of the phase plate is 100% (which maximizes theamplitude of the reflected field E_(r)(r) in the integrals of Eqs.6a-6c), and (4) the phase plate is normal to the incident beam.Conditions (3) and (4) imply that any loss in the optical path willdirectly affect the sensitivity of the sensor. Thus, assuming the phaseplate as having a reflectivity of 100% and that there is little or noloss in the optical path, there are two parameters which determine themaximum sensitivity: well depth and relative diaphragm radius toincident beam size.

In certain embodiments, the optimum operating condition can be foundusing numerical calculations or analytical calculations. Using thenumerical expression of the coupling efficient (Eq. 1), the optimumoperating point can be found, which gives the maximum displacementsensitivity σ. For example, using an operating wavelength of 1550 nm,there are two variables to calculating o: the well depth h₀ and theratio κ=r_(dia)/w_(col), where r_(dia) and w_(col) are the sensordiaphragm radius and the collimated beam waist, respectively. FIG. 10Ais a plot of a numerically calculated displacement sensitivity and FIG.10B is a plot of an analytically calculated displacement sensitivityresulting from varying these two variables. The maximum value of a inFIG. 10A (using the numerical calculations) equals 1.6125×10⁶ m⁻¹ ath₀=193.74 nm and κ=0.64. The maximum value of σ in FIG. 10B (using theanalytical calculations) equals 1.6110×10⁶ m⁻¹ at h₀=193.75 nm andκ=0.64. Both analytical and numerical calculations show that thesensitivity is maximum with h₀=193.75 nm and κ=0.64. FIG. 11Aillustrates the coupling coefficient at a static condition at rest (η₀)as a function of well depth h₀ for a fixed value of κ=0.64. FIG. 11Billustrates the displacement sensitivity σ as a function of well depthh₀ for a fixed value of κ=0.64, with the sensitivity going to zero ath₀=λ/4. FIG. 11C illustrates the displacement sensitivity σ as afunction of wavelength X for h₀=λ/8 and κ=0.64.

Certain embodiments described herein provide a compact fiber sensor thatutilizes for the first time the principle of phase-front modulation todetect acoustic waves at extremely low pressures. In certainembodiments, the sensor can utilize a reflective diaphragm with a π/2phase step microfabricated in a silicon wafer, combined with asingle-mode fiber acting as a spatial filter, to form a simpleinterferometric sensor head. In certain embodiments, the sensor canpresent several advantages over state-of-the-art, high-sensitivity,diaphragm-based, fiber Fabry-Perot sensors. As a two-waveinterferometer, for the same diaphragm dimensions and reflectivity, itcan be less sensitive, but its noise can be also lower in the sameratio, so that its minimum detectable strain (or strain resolution) isnominally the same. Also, because it is a two-wave interferometer, itssensitivity can depend very weakly on the operating wavelength: thelatter can be changed by ±λ/10 for the sensitivity to decrease by ±10%.Unlike previously-developed FP-based fiber acoustic sensors, in certainembodiments, the operating wavelength can be set during themicrofabrication instead of during assembly, and it is therefore muchmore reproducible, in addition of being much less critical. Finally, incertain embodiments, it is much easier and faster to fabricate thesensor head in the clean room and to assemble the sensor.

As described above, a simple analytical expression can be used for thesensitivity of this sensor to acoustic pressure. A laboratory prototypeof an example sensor in accordance with certain embodiments describedherein was fabricated and analyzed to have an average minimum detectablepressure as low as 5.4 μPa/√Hz between 1 kHz and 30 kHz, in agreementwith a theoretical model. Straightforward improvements can be made toimprove this figure down to the 0.2 μPa/√Hz level. In certainembodiments, the sensor has a great potential in various areas,including in vivo pressure monitoring, surveillance, seismic research,structural health monitoring, photoacoustic imaging, stem cell research,and in sensor array networks for oil and gas exploration.

Various embodiments have been described above. Although this inventionhas been described with reference to these specific embodiments, thedescriptions are intended to be illustrative and are not intended to belimiting. Various modifications and applications may occur to thoseskilled in the art without departing from the true spirit and scope ofthe invention as defined in the claims.

What is claimed is:
 1. A sensor comprising: at least one opticalwaveguide configured to emit light in a direction; and an opticalreflector optically coupled to the at least one optical waveguide, theoptical reflector configured to reflect at least a portion of the light,the optical reflector comprising: a first portion of the opticalreflector configured to reflect a first portion of the light back to theat least one optical waveguide; and a second portion of the opticalreflector configured to reflect a second portion of the light back tothe at least one optical waveguide, the second portion of the opticalreflector responsive to a perturbation by moving relative to the firstportion of the optical reflector, the reflected second portion of thelight differing in phase from the reflected first portion of the lightby a phase difference that is not substantially equal to an integermultiple of π when the second portion of the optical reflector is in anequilibrium position in absence of the perturbation.
 2. The sensor ofclaim 1, wherein the at least one optical waveguide comprises an opticalfiber.
 3. The sensor of claim 2, wherein the at least one opticalwaveguide further comprises at least one lens configured to receive thelight from the optical fiber and to transmit the light towards theoptical reflector.
 4. The sensor of claim 3, wherein the at least onelens comprises a graded-index lens.
 5. The sensor of claim 3, whereinthe at least one lens is configured to collimate the light emitted fromthe optical fiber and to return the reflected first portion of the lightand the reflected second portion of the light to the optical fiber. 6.The sensor of claim 1, further comprising a laser configured to generatethe light, the light having at least one wavelength, the sensor furthercomprising an optical signal analyzer configured to receive thereflected first portion of the light and the reflected second portion ofthe light from the at least one optical waveguide.
 7. The sensor ofclaim 1, wherein the optical reflector comprises a wafer, the firstportion of the optical reflector comprising a first surface of thewafer, the second portion of the optical reflector comprising adiaphragm offset from the first portion of the optical reflector alongthe direction, the diaphragm in a well surrounded by the first surface.8. The sensor of claim 7, wherein the diaphragm is thinner than portionsof the wafer surrounding the diaphragm.
 9. The sensor of claim 7,wherein the diaphragm is elastically deformable in response to theperturbation.
 10. The sensor of claim 7, wherein the light has awavelength and the well has a depth substantially equal to one-eighth ofthe wavelength.
 11. The sensor of claim 7, wherein the light has a widthgreater than a width of the well.
 12. The sensor of claim 7, wherein thelight has a width less than or equal to a width of the well.
 13. Thesensor of claim 7, wherein the optical reflector comprises a pluralityof spring structures mechanically coupled to the first portion of theoptical reflector and to the diaphragm, the plurality of springstructures configured to elastically stretch and move such that thediaphragm translates in a direction that is generally perpendicular tothe diaphragm.
 14. The sensor of claim 7, wherein the diaphragm ismechanically coupled to the wafer along a perimeter of the diaphragm,and the diaphragm is configured to elastically bow back and forth in adirection that is generally perpendicular to the diaphragm.
 15. Thesensor of claim 1, wherein the phase difference is substantially equalto an odd integer multiple of π/2.
 16. The sensor of claim 1, whereinthe phase difference has a magnitude that is in at least one of thefollowing ranges: substantially greater than zero and substantially lessthan π, substantially greater than π and substantially less than 2π,substantially greater than 2π and substantially less than 3π,substantially greater than zero and less than or equal to π/2, greaterthan or equal to π/2 and substantially less than π, substantiallygreater than π and less than or equal to 3π/2, greater than or equal to3π/2 and substantially less than 2π, substantially greater than 2π andless than or equal to 5π/2, and greater than or equal to 5π/2 andsubstantially less than 3π.
 17. A method for fabricating a sensor, themethod comprising: providing a first tube comprising a ferrule insidethe first tube, the ferrule comprising an optical waveguide configuredto emit a light beam; inserting at least one lens into the first tube, aportion of the at least one lens extending outwardly past an end of thefirst tube, the at least one lens configured to receive the light beamemitted from the optical waveguide; affixing a first end of a secondtube to the portion of the at least one lens extending outwardly pastthe end of the first tube; and affixing an optical reflector to asurface of a second end of the second tube.
 18. The method of claim 17,further comprising adjusting a distance between the optical waveguide ofthe ferrule and the at least one lens to modify a radius of a collimatedlight beam propagating from the at least one lens.
 19. The method ofclaim 18, further comprising bonding the ferrule and the at least onelens to the first tube.
 20. The method of claim 17, wherein the surfaceof the second end of the second tube is perpendicular to a longitudinalaxis of the second tube within ±0.2 degree.
 21. The method of claim 17,further comprising forming at least one vent hole through a side of thesecond tube.